Multivariate Observations

Note: Supplemental materials are not guaranteed with Rental or Used book purchases.

ISBN: 9780471691211 | 0471691216
Cover: Paperback
Copyright: 8/24/2004

Buy New

Print on Demand: 2-4 Weeks. This item cannot be cancelled or returned.

$196.43

This up-to-date, comprehensive sourcebook treats data-oriented techniques as well as classical methods. Emphasis is on principles rather than mathematical detail, and coverage ranges from the practical problems of graphically representing high dimensional data to the theoretical problems relating to matrices of random variables. Each chapter serves as a self-contained survey of a specific topic. Includes many numerical examples, and over 1,100 references.

NOTATION

xix

1 PRELIMINARIES

1.1 Notation

(2)

1.2 What Is Multivariate Analysis?

(2)

1.3 Expectation and Covariance Operators

(3)

1.4 Sample Data

(2)

1.5 Mahalanobis Distances and Angles

(1)

1.6 Simultaneous Inference

(3)

1.6.1 Simultaneous tests,

(2)

1.6.2 Union-intersection principle,

(1)

1.7 Likelihood Ratio Tests

(1)

Exercises 1,

(3)

2 MULTIVARIATE DISTRIBUTIONS

(42)

2.1 Introduction

(1)

2.2 Multivariate Normal Distribution

(3)

2.3 Wishart Distribution

(8)

2.3.1 Definition and properties,

(2)

2.3.2 Generalized quadratics,

(4)

2.3.3 Noncentral Wishart distribution,

(1)

2.3.4 Eigenvalues of a Wishart matrix,

(1)

2.3.5 Determinant of a Wishart matrix,

(1)

2.4 Hotelling's T² Distribution

(4)

2.4.1 Central distribution,

(4)

2.4.2 Noncentral distribution,

(1)

2.5. Multivariate Beta Distributions

(18)

2.5.1 Derivation,

(3)

2.5.2 Multivariate beta eigenvalues,

(3)

2.5.3 Two trace statistics,

(2)

a. Lawley-Hotelling statistic,

(1)

b. Pillai's trace statistic,

(1)

2.5.4 U-Distribution,

(2)

2.5.5 Summary of special distributions,

(3)

a. Hotelling's T²,

(1)

b. U-statistic,

(1)

c. Maximum root statistic,

(1)

d. Trace statistics,

(1)

e. Equivalence of statistics when mH= 1,

(2)

2.5.6 Factorizations of U.

(14)

a. Product of beta variables,

(3)

b. Product of two U-statistics,

(2)

2.6 Rao's Distribution

(4)

2.7 Multivariate Skewness and Kurtosis

(1)

Exercises 2,

(4)

3 INFERENCE FOR THE MULTIVARIATE NORMAL

(68)

3.1 Introduction

(1)

3.2 Estimation

(4)

3.2.1 Maximum likelihood estimation,

(4)

3.2.2 Distribution theory,

(1)

3.3 Testing for the Mean

(8)

3.3.1 Hoteling's T² test,

(6)

3.3.2 Power of the test,

(1)

3.3.3 Robustness of the test,

(1)

3.3.4 Step-down test procedure.

(1)

3.4 Linear Constraints on the Mean

(15)

3.4.1 Generalization of the paired comparison test,

(1)

3.4.2 Some examples,

(5)

a. Repeated-measurement designs,

(4)

b. Testing specified contrasts,

(1)

c. Test for symmetry,

(1)

d. Testing for a polynomial growth trend,

(1)

e. Independent estimate of

(1)

3.4.3 Minimization technique for the test statistic,

(4)

3.4.4 Confidence intervals,

(4)

3.4.5 Functional relationships (errors in variables regression),

(1)

3.5 Inference for the Dispersion Matrix

(16)

3.5.1 Introduction,

(1)

3.5.2 Blockwise independence: Two blocks,

(3)

a. Likelihood ratio test,

(1)

b. Maximum root test,

(1)

3.5.3 Blockwise independence: b Blocks,

(2)

3.5.4 Diagonal dispersion matrix,

(2)

3.5.5 Equal diagonal blocks,

(1)

3.5.6 Equal correlations and equal variances,

(1)

3.5.7 Simultaneous confidence intervals for correlations,

(2)

3.5.8 Large-sample inferences,

(4)

3.5.9 More general covariance structures,

102

(1)

3.6 Comparing Two Normal Populations

102

(22)

3.6.1 Tests for equal dispersion matrices,

102

(6)

a. Likelihood ratio test,

103

(2)

b. Union-intersection test,

105

(1)

c. Robustness of tests,

106

(1)

d. Robust large-sample tests,

107

(1)

3.6.2 Test for equal means assuming equal dispersion matrices,

108

(6)

a. Hotelling's T² test,

108

(3)

b. Effect of unequal dispersion matrices,

111

(1)

c. Effect of nonnormality,

112

(2)

3.6.3 Test for equal means assuming unequal dispersion matrices,

114

(3)

3.6.4 Profile analysis: Two populations,

117

(7)

Exercises 3,

124

(3)

4 GRAPHICAL AND DATA - ORIENTED TECHNIQUES

127

(48)

4.1 Multivariate Graphical Displays

127

(11)

4.2 Transforming to Normality

138

(3)

4.2.1 Univariate transformations,

138

(2)

4.2.2 Multivariate transformations,

140

(1)

4.3 Distributional Tests and Plots

141

(15)

4.3.1 Investigating marginal distributions,

141

(7)

4.3.2 Tests and plots for multivariate normality,

148

(8)

4.4 Robust Estimation

156

(13)

4.4.1 Why robust estimates?,

156

(1)

4.4.2 Estimation of location,

156

(6)

a Univariate methods,

156

(6)

b. Multivariate methods,

162

(1)

4.4.3 Estimation of dispersion and covariance,

162

(11)

a. Univariate methods,

162

(3)

b. Multivariate methods,

165

(4)

4.5 Outlying Observations

169

(4)

Exercises 4,

173

(2)

5 DIMENSION REDUCTION AND ORDINATION

175

(1)

5.1 Introduction

175

(1)

5.2 Principal Components

176

(103)

5.2.1 Definition,

176

(1)

5.2.2 Dimension reduction properties,

176

(5)

5.2.3 Further properties,

181

(3)

5.2.4 Sample principal components,

184

(13)

5.2.5 Inference for sample components,

197

(3)

5.2.6 Preliminary selection of variables,

200

(1)

5.2.7 General applications,

200

(3)

5.2.8 Generalized principal components analysis,

203

(1)

5.3 Biplots and h-Plots

204

(9)

5.4 Factor Analysis

213

(1)

5.4.1 Underlying model,

213

(3)

5.4.2 Estimation procedures,

216

(6)

a. Method of maximum likelihood,

216

(3)

b. Principal factor analysis,

219

(1)

c. Estimating factor scores,

220

(2)

5.4.3 Some difficulties,

222

(13)

a. Principal factor analysis,

225

(5)

b. Maximum likelihood factor analysis,

230

(1)

c. Conclusions,

231

(4)

5.5 Multidimensional Scaling

235

(46)

5.5.1 Classical (metric) solution,

235

(6)

5.5.2 Noninetric scaling,

241

(12)

5.6 Procrustes Analysis (Matching Configurations)

253

(3)

5.7 Canonical Correlations and Variates

256

(1)

5.7.1 Population correlations,

256

(4)

5.7.2 Sample canonical correlations,

260

(4)

5.7.3 Inference,

264

(3)

5.7.4 More than two sets of variables,

267

(2)

5.8 Discriminant Coordinates

269

(4)

5.9 Assessing Two-Dimensional Representations

273

(1)

5.10 A Brief Comparison of Methods

274

(1)

Exercises 5,

275

(4)

6 DISCRIMINANT ANALYSIS

279

(1)

6.1 Introduction

279

(1)

6.2 Two Groups: Known Distributions

280

(67)

6.2.1 Misclassification errors,

280

(1)

6.2.2 Some allocation principles,

281

(6)

a. Minimize total probability of misclassification,

281

(4)

b. Likelihood ratio method,

285

(1)

c. Minimize total cost of misclassification,

285

(1)

d. Maximize the posterior probability,

285

(1)

e. Minimax allocation,

286

(1)

f. Summary,

287

(1)

6.3 Two Groups: Known Distributions With Unknown Parameters

287

(6)

6.3.1 General methods,

287

(6)

6.3.2 Normal populations,

293

(10)

a. Linear discriminant function,

293

(4)

b. Quadratic discriminant function,

297

(1)

c. Robustness of LDF and QDF,

297

(3)

d. Missing values,

300

(1)

e. Predictive discriminant,

301

(2)

6.3.3 Multivariate discrete distributions,

303

(3)

a. Independent binary variables,

303

(1)

b. Correlated binary variables,

304

(2)

c. Correlated discrete variables,

306

(1)

6.3.4 Multivariate discrete-continuous distributions,

306

(2)

6.4 Two Groups: Logistic Discriminant

308

(1)

6.4.1 General model,

308

(1)

6.4.2 Sampling designs,

309

(3)

a. Conditional sampling,

309

(1)

b. Mixture sampling,

310

(1)

c. Separate sampling,

310

(2)

6.4.3 Computations,

312

(5)

6.4.4 LGD versus LDF,

317

(1)

6.4.5 Predictive logistic model,

318

(1)

6.4.6 Quadratic discrimination,

319

(1)

6.5 Two Groups: Unknown Distributions

320

(1)

6.5.1 Kernel method,

320

(3)

a. Continuous data,

320

(2)

b. Binary data,

322

(1)

c. Continuous and discrete data,

323

(1)

6.5.2 Other nonparametric methods,

323

(14)

a Nearest neighbor techniques,

323

(1)

b. Partitioning methods,

324

(1)

c. Distance methods,

324

(1)

d. Rank procedures,

324

(2)

e. Sequential discrimination on the variables,

326

(1)

6.6 Utilizing Unclassified Observations

327

(1)

6.7 All Observations Unclassified (Method of Mixtures)

328

(1)

6.8 Cases of Doubt

329

(1)

6.9 More Than Two Groups

330

(7)

6.10 Selection of Variables

337

(5)

6.10.1 Two groups,

337

(4)

6.10.2 More than two groups,

341

(1)

6.11 Some Conclusions

342

(1)

Exercises 6,

343

(4)

7 Curs FFtt ANALYSIS

347

(48)

7.1 Introduction

347

(4)

7.2 Proximity data

351

(8)

7.2.1 Dissimilarities,

351

(5)

7.2.2 Similarities,

356

(3)

7.3 Hierarchical Clustering: Agglomerative Techniques

359

(17)

7.3.1 Some commonly used methods,

360

(8)

a. Single linkage (nearest neighbor) method,

360

(1)

b. Complete linkage (farthest neighbor) method,

361

(1)

c. Centroid method,

362

(1)

d. Incremental sum of squares method,

363

(1)

e. Median method,

363

(1)

f. Group average method,

363

(1)

g. Lance and Williams flexible method,

364

(1)

h. Information measures,

365

(1)

i. Other methods,

366

(2)

7.3.2 Comparison of methods,

368

(8)

a. Monotonicity,

368

(5)

b. Spatial properties,

373

(2)

c. Computational effort,

375

(1)

7.4 Hierarchical Clustering: Divisive Techniques

376

(3)

7.4.1 Monothetic methods.,

377

(1)

7.4.2 Polvthetic methods,

378

(1)

7.5. Partitioning Methods

379

(8)

7.5.1 Number of partitions,

379

(1)

7.5.2 Starling the process,

379

(1)

7.5.3 Reassigning the objects,

380

(6)

a. Minimize trace W,

382

(1)

b. Minimize \W\,

382

(1)

c. Maximize trace BW-¹,

383

(3)

7.5.4 Other techniques,

386

(1)

7.6 Overlapping Clusters (Clumping)

387

(1)

7.7 Choosing the Number of Clusters

388

(2)

7.8 General Comments

390

(2)

Exercises 7,

392

(3)

8 MULTIVARIATE LINEAR MODELS

395

(38)

8.1 Least Squares Estimation

395

(4)

8.2 Properties of Least Squares Estimates

399

(4)

8.3 Least Squares With Linear Constraints

403

(2)

8.4 Distribution Theory

405

(3)

8.5 Analysis of Residuals

408

(1)

8.6 Hypothesis Testing

409

(14)

8.6.1 Extension of univariate theory,

409

(2)

8.6.2 Test procedures,

411

(6)

a. Union-intersection test,

411

(1)

b. Likelihood ratio test,

412

(1)

c. Other test statistics,

413

(1)

d. Comparison of four test statistics,

414

(2)

e. Mardia's permutation test,

416

(1)

8.6.3 Simultaneous confidence intervals,

417

(2)

8.6.4 Comparing two populations,

419

(2)

8.6.5 Canonical form,

421

(1)

8.6.6 Missing observations,

422

(1)

8.6.7 Other topics,

423

(1)

8.7 A Generalized Linear Hypothesis

423

(3)

8.7.1 Theory,

423

(1)

8.7.2 Profile analysis for Kappa populations,

424

(1)

8.7.3 Tests for mean of multivariate normal,

425

(1)

8.8 Step-Down Procedures

426

(2)

8.9 Multiple Design Models

428

(3)

Exercises 8,

431

(2)

9 MULTIVARIATE ANALYSIS OF VARIANCE AND COVARIANCE

433

(63)

9.1 Introduction

433

(1)

9.2 One-Way Classification

433

(21)

9.2.1 Hypothesis testing,

433

(4)

9.2.2 Multiple comparisons,

437

(3)

9.2.3 Comparison of test statistics,

440

(3)

9.2.4 Robust tests for equal means,

443

(4)

a. Permutation test,

443

(2)

b. James' test,

445

(2)

9.2.5 Reparameterization,

447

(1)

9.2.6 Comparing dispersion matrices,

448

(4)

a. Test for equal dispersion matrices,

448

(3)

b. Graphical comparisons,

451

(1)

9.2.7 Exact procedures for means assuming unequal dispersion matrices,

452

(2)

a. Hypothesis testing,

452

(2)

b. Multiple confidence intervals,

454

(1)

9.3 Randomized Block Design

454

(4)

9.3.1 Hypothesis testing and confidence intervals,

454

(4)

9.3.2 Underlying assumptions,

458

(1)

9.4 Two-Way Classification With Equal Observations per Mean

458

(5)

9.5 Analysis of Covariance

463

(9)

9.5.1 Univariate theory,

463

(2)

9.5.2 Multivariate theory,

465

(6)

9.5.3 Test for additional information,

471

(1)

9.6 Multivariate Cochran's Theorem on Quadratics

472

(2)

9.7 Growth Curve Analysis

474

(20)

9.7.1 Examples,

474

(4)

a. Single growth curve,

474

(1)

b. Two growth curves,

475

(1)

c. Single growth curve for a randomized block design,

476

(1)

d. Two-dimensional growth curve,

477

(1)

9.7.2 General theory,

478

(6)

a. Potthoff and Roy's method,

479

(1)

b. Rao Khatri analysis of covariance method,

480

(3)

c. Choice of method,

483

(1)

9.7.3 Single growth curve,

484

(2)

9.7.4 Test for internal adequacy,

486

(1)

9.7.5 A case study,

487

(5)

9.7.6 Further topics,

492

(2)

Exercises 9,

494

(2)

10 SPECIAI, TOPICS

496

(21)

10.1 Computational Techniques

496

(16)

10.1.1 Solving the normal equations,

496

(2)

a. Cholesky decomposition,

496

(1)

b. QR-algorithm,

497

(1)

10.1.2 Hypothesis matrix,

498

(1)

10.1.3 Calculating Hotelling's T²,

499

(1)

10.1.4 Generalized symmetric eigenproblem,

500

(4)

a. Multivariate linear hypothesis,

501

(2)

b. One-way classification,

503

(1)

c. Discriminant coordinates,

504

(1)

10.1.5 Singular value decomposition,

504

(3)

a. Definition,

504

(1)

b. Solution of normal equations,

505

(1)

c. Principal components,

506

(1)

d. Canonical correlations,

506

(1)

10.1.6 Selecting the best subset,

507

(24)

a. Response variables,

507

(3)

b. Regressor variables,

510

(2)

10.2 Log-Linear Models for Binary Data

512

(2)

10.3 Incomplete Data

514

(3)

APPENDIX

517

(98)

A Some Matrix Algebra

517

(1)

A1 Trace and eigenvalues,

517

(1)

A2 Rank,

518

(1)

A3 Patterned matrices,

519

(2)

A4 Positive semidefinite matrices,

521

(1)

A5 Positive definite matrices,

521

(1)

A6 Idempotent matrices,

522

(1)

A7 Optimization and inequalities,

523

(7)

A8 Vector and matrix differentiation,

530

(1)

A9 Jacobians and transformations,

531

(1)

A10 Asymptotic normality,

532

(1)

B Orthogonal Projections

533

(6)

B1 Orthogonal decomposition of vectors,

533

(1)

B2 Orthogonal complements,

534

(1)

B3 Projections on subspaces,

535

(4)

C Order Statistics and Probability Plotting

539

(6)

C1 Sample distribution functions,

539

(1)

C2 Gamma distribution,

540

(1)

C3 Beta distribution,

541

(1)

C4 Probability plotting,

542

(3)

D Statistical Tables

545

(1)

D1 Bonferroni t -percentage points,

546

(4)

D2 Maximum likelihood estimates for the gamma distribution,

550

(1)

D3 Upper tail percentage points for bl,

551

(1)

D4 Coefficients in a normalizing transformation of b1,

551

(2)

D5 Simulation percentiles for b2,

553

(1)

D6 Charts for the percentiles of b2,

554

(2)

D7 Coefficients for the Wilk-Shapiro (W) test,

556

(2)

D8 Percentiles for the Wilk-Shapiro (W) test,

558

(1)

D9 D'Agostino's test for normality,

558

(2)

D10 Anderson-Darling (A²n) test for normality,

560

(1)

Dl1 Discordancy test for single gamma outlier,

561

(1)

D12 Discordancy test for single multivariate normal outlier,

562

(1)

D13 Wilks' likelihood ratio test,

562

(1)

D14 Roy's maximum root statistic,

563

(1)

D15 Lawley-Hotelling trace statistic,

563

(1)

D16 Pillai's trace statistic,

564

(1)

D17 Test for mutual independence,

564

(1)

D18 Test for equal dispersion matrices with equal sample sizes,

564

(51)

OUTLINE SOLUTIONS TO EXERCISES

615

(11)

REFERENCES

626

(45)

INDEX

671

Please wait while the item is added to your bag...

FREE SHIPPING

Amazon no longer offers textbook rentals. We do!

Multivariate Observations

Summary

Author Biography

Table of Contents

Supplemental Materials